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A computer search shows that the set of initial 70 Mersenne numbers $2^p-1$ (all with prime exponents) has 54 members, or approx 77%, that lie between twin practical numbers. (Edit: Alternatively, $2^p-2$ is a practical number 77% of the time for the first 70 prime $p$.) Is it known whether this percentage will increase, decrease, or converge to a limit as the sample size tends to infinity?

A similar search shows that for the first 13 Mersenne primes $2^p-1$, then $2^p-2$ is a practical number with one exception, namely $p=107$. Are there any other exceptions? My computing power is unable to cope with larger Mersenne primes.

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    $\begingroup$ Looks like an instance of the law of small numbers to me. The density of practical numbers is not too small, and if $p$ is prime, then $2^p-2$ is divisible by 2 and 3, and the probability of being divisible by a given small prime is significantly larger than for a random integer. So I would believe that the observed pattern will vanish from some point onwards. $\endgroup$ Dec 26, 2015 at 21:23
  • $\begingroup$ @Frank: The triplet is $2^n-2,\, 2^n-1,\, 2^n$. Since $2^n$ is automatically a practical number, then you can greatly simplify your question by just focusing on $2^p-2$, for prime $p$. (P.S. Are you sure $2^{107}-2$ is not a practical number? It seems to satisfy the Stewart-Sierpienski condition given in the link above.) $\endgroup$ Jan 2, 2016 at 4:48
  • $\begingroup$ @Tito: Thank you for the clarifying edit. Wolfram|Alpha gives the factors of $2^107-2$ as 2×3×107×6361×69431×20394401×28059810762433 (7 distinct prime factors). The first 5 lowest divisors are 1, 2, 3, 6, 107 and this is not a [complete sequence][1]. [1]: en.wikipedia.org/wiki/Complete_sequence $\endgroup$ Jan 2, 2016 at 8:36
  • $\begingroup$ @Jan-Christoph: If $p$ is prime, then $2^p-2$ is divisible by 2, 3 (as you have noted) and by $p$ (using Fermat's little theorem). The latter fact has enabled me to partially factorize some of the larger $2^p-2$ numbers to see if they are not practical. I believe that the $2^p-2$ for $p$ (an exponent of a Mersenne prime) in the following set are not practical {107, 607, 2203, 4253, 11213, 86243, 756839} This set is not exhaustive. $\endgroup$ Jan 2, 2016 at 15:28
  • $\begingroup$ @Franz: Can this question that has been edited several times for clarity have its closed status undone? What do you advise as the topic has shown some interest? $\endgroup$ Jan 2, 2016 at 16:51

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